Problem 20 - A Graphical Approach to College Algebra

Question

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=12 x^{107,499}$$

Step-by-Step Solution

Answer

The end behavior shows the graph rising right and falling left.

Step 1: Identify the Leading Term and Its Degree

Since the polynomial given is \( P(x) = 12x^{107,499} \), we note that the leading term is \( 12x^{107,499} \). This term determines the end behavior of the function. The degree of the polynomial is \( 107,499 \), which is an odd number.

Step 2: Determine the Leading Coefficient

The leading coefficient is the coefficient of the leading term, which in this case is \( 12 \). Since \( 12 \) is a positive number, it will affect the direction of the ends of the polynomial.

Step 3: Determine the End Behavior Based on Degree and Leading Coefficient

For polynomials with odd degrees, the end behavior is determined by the sign of the leading coefficient. Since the degree is odd and the leading coefficient \( 12 \) is positive, the graph of the polynomial will rise to the right and fall to the left.

Step 4: Draw an End Behavior Diagram

Visualize the end behavior of the function by drawing a diagram. The diagram will show the graph beginning low on the left and rising high on the right: End behavior: as \( x \rightarrow -\infty \), \( P(x) \rightarrow -\infty \) and as \( x \rightarrow +\infty \), \( P(x) \rightarrow +\infty \).

Key Concepts

Leading Term

The leading term of a polynomial is the term with the highest power of the variable, here represented as "x". In our exercise, the polynomial is given by \( P(x) = 12x^{107,499} \). This means that the leading term is \( 12x^{107,499} \). The leading term plays a crucial role because it dictates the behavior of the polynomial graph, especially as \( x \) becomes very large or very small.

Understanding the leading term helps in analyzing how the polynomial behaves, particularly at the ends. The power of the leading term suggests whether the polynomial's graph will have symmetry or how the tails of the graph act. When you're determining end behavior, remember, the leading term holds the key.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. In the given polynomial \( P(x) = 12x^{107,499} \), the degree is 107,499. This is a very high degree, and notably, it's an odd number. The degree of a polynomial gives valuable insight into the nature of its graph.

• If the degree is odd, as in this polynomial, the graph will have opposite end behaviors on either side.
• If the degree were even, the graph would end in the same direction on both sides, either both going up or both going down.

In simple terms, the degree can tell us whether the ends of the graph will move "together" or "apart" as the value of \( x \) approaches the extremes. Hence, knowing the degree is critical for anticipating how the graph behaves on a large scale.

Leading Coefficient

The leading coefficient is the number in front of the variable in the leading term. For the polynomial \( P(x) = 12x^{107,499} \), the leading coefficient is 12. This coefficient affects the direction in which the ends of the polynomial's graph point.

• A positive leading coefficient means that as \( x \rightarrow +\infty \), \( P(x) \rightarrow +\infty \).
• On the other hand, a negative leading coefficient would cause the graph to go downwards as \( x \) increases.

The sign of the leading coefficient, combined with the degree, helps determine the graph's end behavior. For odd degrees, a positive leading coefficient leads to the graph ending upwards on the right and downwards on the left. Understanding these interactions aids enormously in sketching and interpreting polynomial graphs without even using a calculator.